Special topics

Fixed effects When trying to compare apples with apples, we worry

about the numerous potential differences on confounding variables

If differences on confounding variables are stable over time, we can eliminate bias from them by only analyzing variation within the same unit over time E.g., breast-feeding study (unit is woman) E.g., income and partisanship (unit is state)

To only analyze variation within the same unit over time, we use fixed effects Stata commands areg and xtreg Equivalent to adding indicator (or dummy variables) variables for

units Equivalent to between subjects design (as opposed to within

subjects)

Inter-State Variation

xtreg [DV] [IV], be

(or)

collapse [DV] [IV], by(state)reg [DV] [IV]

Intra-State Variation

xtreg [DV] [IV], fe

(or)

areg [DV] [IV], a(state)

Aggregate Panel Variation

reg [DV] [IV]

Intra- or Inter-country Variation?(animated slide, see summary on next slide)

DV

IV

e.g., income

e.g., partisanship (Dem

ocrat %)

Intra- or Inter-State(Person/Firm) Variation?

DV

IV

Aggregate Panel Variation

reg [DV] [IV]

DV

IV

Fixed effects (fe)Intra-State Variation

xtreg [DV] [IV], fe (or)

areg [DV] [IV], a(state)

DV

IV

Between effects (be)Inter-State Variation

xtreg [DV] [IV], be(or)

collapse [DV] [IV], by(state)reg [DV] [IV]

Fixed effects

ProblemsThrows away potentially relevant variation

(alternative: random effects)Variation over time may be primarily from

random measurement error (e.g., unions and wages)

Unusual factors may drive changes in explanatory variables over time and also influence the dependent variable (e.g., currency unions)

Interactions

Interactions Interactions test whether the combination of variables affects the

outcome differently than the sum of the main (or individual) effects. Examples

Interaction between adding sugar to coffee and stirring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does.

Interaction between smoking and inhaling asbestos fibres: Both raise lung carcinoma risk, but exposure to asbestos multiplies the cancer risk in smokers.

Example from problem set How would we test whether defendants are sentenced to death more

frequently when their victims are both white and strangers than you would expect from the coefficients on white victim and on victim stranger.

Interactions . g wvXvs = wv* vs. reg death bd yv ac fv v2 ms wv vs wvXvs ----------------------------------------------- death | Coef. Std. Err. t P>|t|-------+--------------------------------------- (omitted) wv | .0985493 .1873771 0.53 0.600 vs | .1076086 .2004193 0.54 0.593 wvXvs | .3303334 .2299526 1.44 0.154 _cons | .0558568 .2150039 0.26 0.796-----------------------------------------------

•To interpret interactions, substitute the appropriate values for each variable•E.g., what’s the effect for

• .099 wv+.108 vs+.330 wvXvs •White, non-stranger: .099(1)+.108(0)+.330(1)*(0) = .099•White, stranger: .099(1)+.108(1)+.330(1)*(1) = .537•Black, non-stranger: .099(0)+.108(0)+.330(0)*(0) = comparison•Black, stranger: .099(0)+.108(1)+.330(1)*(0) = .108

Interactions

. tab wv vs, sum(death)

Means, Standard Deviations and Frequencies of death

| vs wv | 0 1 | Total-----------+----------------------+---------- 0 | .16666667 .28571429 | .23076923 | .38924947 .46880723 | .42966892 | 12 14 | 26-----------+----------------------+---------- 1 | .40540541 .75675676 | .58108108 | .49774265 .43495884 | .4967499 | 37 37 | 74-----------+----------------------+---------- Total | .34693878 .62745098 | .49 | .48092881 .48829435 | .50241839 | 49 51 | 100

Importance of a variable

Death penalty example

. sum death bd- yv

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- death | 100 .49 .5024184 0 1 bd | 100 .53 .5016136 0 1 wv | 100 .74 .440844 0 1 ac | 100 .4366667 .225705 0 1 fv | 100 .31 .4648232 0 1-------------+-------------------------------------------------------- vs | 100 .51 .5024184 0 1 v2 | 100 .14 .3487351 0 1 ms | 100 .12 .3265986 0 1 yv | 100 .08 .2726599 0 1

Death penalty example

. reg death bd-yv , beta------------------------------------------------ death | Coef. Std. Err. P>|t| Beta-------+----------------------------------------- bd | -.0869168 .1102374 0.432 -.0867775 wv | .3052246 .1207463 0.013 .2678175 ac | .4071931 .2228501 0.071 .1829263 fv | .0790273 .1061283 0.458 .0731138 vs | .3563889 .101464 0.001 .3563889 v2 | .0499414 .1394044 0.721 .0346649 ms | .2836468 .1517671 0.065 .1843855 yv | .050356 .1773002 0.777 .027328 _cons | -.1189227 .1782999 0.506 .-------------------------------------------------

Importance of a variable

Three potential answersTheoretical importanceLevel importanceDispersion importance

Importance of a variable

Theoretical importanceTheoretical importance = Regression

coefficient (b) To compare explanatory variables, put them

on the same scale E.g., vary between 0 and 1

Importance of a variable

Level importance: most important in particular times and placesE.g., did the economy or presidential

popularity matter more in congressional races in 2006?

Level importance= bj* xj

Importance of a variable

Dispersion importance: what explains the variance on the dependent variable E.g., given that the GOP won in this particular

election, why did some people vote for them and others against?

Dispersion importance = Standardized coefficients, or alternatively Regression coefficient times standard deviation of

explanatory variable In bivariate case, correlation

Which to use?

Depends on the research questionUsually theoretical importanceSometimes level importanceDispersion importance not usually relevant

Partial residual scatter plots

Partial residual scatter plots

Importance of plotting your data Importance of controls How do you plot your data after you’ve

accounted for control variables? Example: inferences about candidates in

Mexico from faces

MIT students rated Mexican candidates faces on “competence”

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Regression Source | SS df MS Number of obs = 33-------------+------------------------------ F( 3, 29) = 4.16 Model | .082003892 3 .027334631 Prob > F = 0.0144 Residual | .190333473 29 .006563223 R-squared = 0.3011-------------+------------------------------ Adj R-squared = 0.2288 Total | .272337365 32 .008510543 Root MSE = .08101

------------------------------------------------------------------------------ vote_a | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- competent | .1669117 .0863812 1.93 0.063 -.0097577 .343581 incumbent | .0110896 .0310549 0.36 0.724 -.0524248 .074604 party_a | .2116774 .1098925 1.93 0.064 -.013078 .4364327 _cons | .2859541 .0635944 4.50 0.000 .1558889 .4160194------------------------------------------------------------------------------

vote_a is vote share for Candidate A incumbent is a dummy variable for whether the party currently holds the office party_a is the vote share for the party of Candidate A in the previous election

We want to create a scatter plot of vote_a by competent controlling for incumbent and party_a

Reminder:

Residuals

ei = Yi – B0 – B1Xi

Calculating partial residuals

First run your regression with all the relevant variables. reg vote_a competent incumbent party_a

To calculate the residual for the full model, use. predict e, res(This creates a new variable “e”, which equals to the residual.)

Here, however, we want to generate the residual controlling only for some variables. To do this, we could manually predict vote_a based only on incumbent and party_a:

. g y_hat = (0)*.167+ incumbent*.011 + party_a*.212

We can then generate the partial residual. g partial_e = vote_a – y_hat

Instead, can use the Stata adjust. adjust competent = 0, by(incumbent party_a) gen(y_hat). g partial_e = vote_a – y_hat

Source | SS df MS Number of obs = 33-------------+------------------------------ F( 3, 29) = 4.16

Model | .082003892 3 .027334631 Prob > F = 0.0144Residual | .190333473 29 .006563223 R-squared = 0.3011

-------------+------------------------------ Adj R-squared = 0.2288Total | .272337365 32 .008510543 Root MSE = .08101

------------------------------------------------------------------------------vote_a | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------competent | .1669117 .0863812 1.93 0.063 -.0097577 .343581incumbent | .0110896 .0310549 0.36 0.724 -.0524248 .074604

party_a | .2116774 .1098925 1.93 0.064 -.013078 .4364327_cons | .2859541 .0635944 4.50 0.000 .1558889 .4160194

------------------------------------------------------------------------------

vote_a is vote share for Candidate A incumbent is a dummy variable for whether the party currently holds the office party_a is the vote share for the party of Candidate A in the previous election

We want to create a scatter plot of vote_a by competent controlling for incumbent and party_a

Calculating partial residuals

Regression of the partial residual on competent. (Should not necessarily give you the same coefficient estimate because competent is not residualized.)

. reg partial_e competent Source | SS df MS Number of obs = 33-------------+------------------------------ F( 1, 31) = 4.52 Model | .027767147 1 .027767147 Prob > F = 0.0415 Residual | .190333468 31 .006139789 R-squared = 0.1273-------------+------------------------------ Adj R-squared = 0.0992 Total | .218100616 32 .006815644 Root MSE = .07836

------------------------------------------------------------------------------ e | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- competent | .1669117 .078487 2.13 0.042 .0068364 .326987 _cons | -7.25e-09 .0470166 -0.00 1.000 -.0958909 .0958909------------------------------------------------------------------------------

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Compare scatter plot (top) with residual scatter plot (bottom)

Residual plots especially important if results change when adding controls

Avplot & cprplot

You can also use avplot to generate residual scatter plots

After you run your regression, use the following command avplot competent

Unlike the method above, avplot also conditions (residualizes) your explanatory variable

Good for detecting outliers Bad for detecting functional form For functional form, use

cprplot, which does not residualize the explanatory variable

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-.4 -.2 0 .2 .4e( competent | X )

coef = .16691168, se = .08638117, t = 1.93

Imputing missing data (on controls)

Imputing missing data

Variables often have missing data Sources of missing data Missing data

May reduce estimate precision (wider confidence intervals b/c smaller sample)

May bias estimates if data is not missing a random To rescue data with missing cases on control

variables: impute using other variables Imputing data can

Increase sample size and so increase precision of estimates

Reduce bias if data is not missing at random

Imputation example

Car ownership in 1948 Say that some percentage of sample

forgot to answer a question about whether they own a car

The data set contains variables that predict car ownership: family_income, family_size, rural, urban, employed

Stata imputation command

impute depvar varlist [weight] [if exp] [in range], generate(newvar1) depvar is the variable whose missing values are to be imputed. varlist is the list of variables on which the imputations are to be

based newvar1 is the new variable to contain the imputations

Example impute own_car family_income family_size rural suburban employed, g(i_own_car)

Rules about imputing

Before you estimate a regression model, use the summary command to check for missing data

Before you impute, check that relevant variables actually predict the variable with missing values (use regression or other estimator)

Don’t use your studies’ dependent variable or key explanatory variable to make the imputation’s (exceptions) Use demographic variables Use variables exogenous to the dependent variable or key

explanatory variables Don’t impute missing values on your studies’ dependent

variable or key explanatory variable (exceptions) Always note whether imputation changed results If too much data is missing, imputation won’t help